Quantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation

Understand

We develop a quantum filter diagonalization method (QFD) that lies somewhere between the variational quantum eigensolver (VQE) and the phase estimation algorithm (PEA) in terms of required quantum circuit resources and conceptual simplicity.

• QFD uses a set of of time-propagated guess states as a variational basis for approximate diagonalization of a sparse Pauli Hamiltonian.
• The variational coefficients of the basis functions are determined by the Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem in the space of time-propagated guess states.
• The matrix elements of the subspace Hamiltonian and subspace metric matrix are each determined in quantum circuits by a one-ancilla extended swap test, i.e., statistical convergence of a one-ancilla PEA circuit.